Nearly Ramanujan conjecture for the graphs A(n, q)

Let $A(n,q)$ be the algebraically defined graph introduced in the paper, and let its second largest eigenvalue mean the second largest eigenvalue of its adjacency matrix. Nearly Ramanujan conjecture. The second largest eigenvalue of $A(n,q)$ is less than or equal to $2\sqrt{q}$. This is motivated by Ustimenko's conjecture for the graphs $D(n,q)$; the source states that the corresponding bound for $A(n,q)$ would follow from that conjecture, while only certain low-dimensional cases are known.